Sauté onions in butter and oil over medium-low until soft and golden, 20–25 minutes; stir in brown sugar, salt, and pepper.
Cook bacon until crisp, drain, and crumble. Let both cool slightly.
Roll pastry to a 10 x 10-inch square, spread onions evenly, sprinkle bacon and Parmesan.
Fold pastry in half, press gently, then cut into 12 strips. Twist each strip several times and place on a parchment-lined sheet.
Brush with egg wash and bake at 400°F (200°C) for 12–15 minutes until puffed and golden. Cool 5 minutes before serving.